Thesis for degree: Licentiate of Engineering

streaming [3]. The concentration ρ ζ is defined similarly as the fluid density in equation (2.7)
[3]:
(2.13)
To avoid numerical instability, it is recommended to keep the relaxation times at the order of
unity. This would mean that the diffusivity values are adjusted by scaling them up to ensure
relaxation times of unity [14-16].
The LB equation can be extended to include several components or species. The equations
remain the same for each species but the interaction and combination of the streaming and
collision action for species i is defined as [3, 14-16].
(2.14)
where f i a is the PDF, e i a the velocity and
time t along the direction a for species i.
a i the collision term at any spatial location x and
Streaming is described in two steps. The PDFs, for example for species 1, are streamed from
one lattice point to the adjacent lattice points, while PDFs for the remaining species, i.e.,
species 2 and 3, are streamed from one lattice point to off-lattice points in the first step. Offlattice
points are defined as sites which are not related to the current start point. In the second
step, the PDF values for species 2 and 3 are determined by interpolation at lattice points. The
collision concept is divided into self-collision, i.e., collision between particles of the same
species and cross-collision, i.e., collision between particles of different species when the
relative velocity between particles of the different species is non-zero [14-16].
For the porous media, the collision term is considered as a second intermediate step after the
streaming [3]. Finally if there are multiple species, the LB model is adjusted by upgrading the
distribution function by also including a composite velocity:
(2.15)
where τ is the relaxation time and i represents the different components involved [3].
To form a realistic model, it is possible to include an external force, e.g., gravitational force,
on the particles or interaction forces between the particles. This force is incorporated in a
velocity term which is added to the velocity calculation. These parts are defined as [3]:
(2.16)
(2.17)
where F is the force acting on the particle, m the mass and a the acceleration of the particle.
Further, u is the velocity, ρ the density and τ the relaxation time.
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